Lemma 1 ([1]})  
If the discrete-state Markov chain \(q(t)\)  is ergodic with stationary distribution \({\pi }\)  and we can find a non-negative solution of \( \lbrace {\mathbf {v}}_{{q}}, {q} \in \mathcal {Q} \rbrace \)  such that
\({\mathbf {v}}_{{q}} \sum _{l \in L_{{q}}}^{} \lambda {(l)}=\mathbf {b}_{{q}} {\pi }_{{q}} + \sum _{l \in L^\prime _{{q}}}^{}\lambda {(l)} {\mathbf {v}}_{q_l} A_{l}, \quad {q} \in \mathcal {Q},\) 
